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Effects of curvature in two - dimensional condensed matter systems

Silva, Ygor Pará

Published Aug 23, 2022• 0 citations
OpenAlex

Abstract

The influence of curved background on the properties of Condensed Matter systems is a significant question for a broad class of physical applications, catching the curiosity because of their interesting features in a rich interface between many research areas. In this thesis, we introduce an analytically solvable model of Dirac fermions with imaginary mass on the sphere. We show the existence of an in nite sequence of exceptional points (EP), which depend on the radius (curvature) of the sphere. We employ quench dynamics to characterize curvaturedependent Non-Hermitian phase transitions. We demonstrate that the existence of singular points of the Loschmidt echo and the fidelity are an unambiguous signature of geometric EPs that distinguish between different phases of the model. Also, we use numerical techniques to solve the Bose-Hubbard hamiltonian that simulates the hyperbolic geometry. In this system, we attempt to verify whether we can observe a bound-states comprising a pair of atoms in the presence of repulsive interactions in a hyperbolic lattice. This system is a playground to simulate a space with negative curvature, that is difficult to simulate in a euclidean space without distortions.

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The influence of curved background on the properties of Condensed Matter systems is a significant question for a broad class of physical applications, catching the curiosity because of their interesting features in a rich interface between many research areas. In this thesis, we introduce an analytically solvable model of Dirac fermions with imaginary mass on the sphere. We show the existence of an in nite sequence of exceptional points (EP), which depend on the radius (curvature) of the sphere. We employ quench dynamics to characterize curvaturedependent Non-Hermitian phase transitions. We demonstrate that the existence of singular points of the Loschmidt echo and the fidelity are an unambiguous signature of geometric EPs that distinguish between different phases of the model. Also, we use numerical techniques to solve the Bose-Hubbard hamiltonian that simulates the hyperbolic geometry. In this system, we attempt to verify whether we can observe a bound-states comprising a pair of atoms in the presence of repulsive interactions in a hyperbolic lattice. This system is a playground to simulate a space with negative curvature, that is difficult to simulate in a euclidean space without distortions.
Non-Hermitian Hamiltonians are relevant to describe the features of a broad class of physical phenomena, ranging from photonics and atomic and molecular systems to nuclear physics and mesoscopic electronic systems. An important question relies on the understanding of the influence of a curved background on the static and dynamical properties of non-Hermitian systems. In this work, we study the interplay of geometry and non-Hermitian dynamics by unveiling the existence of curvature-dependent non-Hermitian phase transitions. We investigate a prototypical model of Dirac fermions on a sphere with an imaginary mass term. This exactly solvable model admits an infinite set of curvature-dependent pseudo-Landau levels. We characterize these phases by computing an order parameter given by the pseudomagnetization and, independently, the non-Hermitian fidelity susceptibility. Finally, we probe the non-Hermitian phase transitions by computing the (generalized) Loschmidt echo and the dynamical fidelity after a quantum quench of the imaginary mass and find singularities corresponding to exceptional radii of the sphere.
One of the challenging problems in the condensed matter physics is to understand the quantum many-body systems, especially, their physical mechanisms behind. Since there are only a few complete analytical solutions of these systems, several numerical simulation methods have been proposed in recent years. Amongst all of them, the Tensor Network algorithms have become increasingly popular in recent years, especially for their adaptability to simulate strongly correlated systems. The current work focuses on the generalization of such Tensor-Network-based algorithms, which are sufficiently robust to describe critical phenomena and phase transitions of multistate spin Hamiltonians in the thermodynamic limit. We have chosen two algorithms: the Corner Transfer Matrix Renormalization Group and the Higher-Order Tensor Renormalization Group. This work, based on tensor-network analysis, opens doors for the understanding of phase transition and entanglement of the interacting systems on the non-Euclidean geometries. We focus on three main topics: A new thermodynamic model of social influence, free energy is analyzed to classify the phase transitions on an infinite set of the negatively curved geometries where a relation between the free energy and the Gaussian radius of the curvature is conjectured, a unique tensor-based algorithm is proposed to study the phase transition on fractal structures.
Dirac exceptional points (EPs) represent a novel class of non-Hermitian singularities that, unlike conventional EPs, reside entirely within the parity-time unbroken phase and exhibit linear energy dispersion. Here, we theoretically investigate the quantum geometry of Dirac EPs realized in nitrogen-vacancy centers in diamond, utilizing fidelity susceptibility as a probe. We demonstrate that despite the absence of a symmetry-breaking phase transition, the Dirac EP induces a pronounced geometric singularity, confirming the validity of the fidelity in characterizing non-Hermitian EPs. Specifically, the real part of the fidelity susceptibility diverges to negative infinity, which serves as a signature of non-Hermitian criticality. Crucially, however, we reveal that this divergence exhibits a distinct anisotropy, diverging along the non-reciprocal coupling direction while remaining finite along the detuning axis. Furthermore, we establish that this anisotropy, characterized by at least one exact dark direction coexisting with divergent directions, is a generic consequence of the Dirac EP structure whenever the parameter derivatives collectively span the off-diagonal operator space at the Dirac EP. This behavior stands in stark contrast to the omnidirectional divergence observed in conventional EPs. Our findings provide a comprehensive picture of the fidelity probe near the Dirac EP, highlighting the critical role of parameter directionality in exploiting Dirac EPs for quantum control and sensing applications.
In this thesis, I will first derive and study the effective field theories of isotropic-nematic quantum phase transitions of Quantum states. The low-energy theory of the nematic field has z=2 dynamics due to a Berry phase of the order parameter, which is related to the Hall viscosity in parity and time-reversal-symmetry (TRS) broken states. The vortex of the nematic field, which is physically a disclination, creates a nonzero geometry curvature in the disclination core. The leading coupling between the nematic field and gauge field includes a Wen-Zee term which links the geometry curvature with the gauge theory. In the second part of this thesis, I investigate the geometry related issues in Weyl semimetals and SPT states, and explore the novel character of geometry defect in SPT states inherited from the topological nature of manybody system. In addition, I would introduce a general way to induce topological phase transition via decorated defect condensate. In the final part of this thesis, I begin with the bilayer Half-filled Landau Level system where the two composite Fermi surface acquires interlayer coherence and forms bonding/anti-bonding composite fermi sea. The corresponding interlayer coherent composite Fermi liquid(ICCFL) phase provides a straightforward landscape to verify the Dirac nature in Son's theory and extract the hidden Berry phase structure of the composite Fermi surface. The ICCFL phase contains two Fermi surfaces which are detached in most regions but adhesive at two hot spots. Such nematic structure is a consequence of the Berry phase encoded in the Dirac Fermi surface which is absent in HLR theory. Due to the nematicity in ICCFL, the system supports half-quantum vortex with deconfined π2\frac{\pi}{2}2π​ gauge flux and the phase transition toward ICCFL contains a Lifshitz criticality with z=3z=3z=3 dynamical exponent. In addition, the exciton order parameter carries topological spin number so the ICCFL contains a unique Wen-Zee term which connects EM response with the background geometry curvature.
In this work, we explore the PT\mathcal{PT}PT-symmetric quantum Rabi model, which describes a PT\mathcal{PT}PT-symmetric qubit coupled to a quantized light field. By employing the adiabatic approximation (AA), we are able to solve this model analytically in the parameter regime of interest and analyze various physical aspects. We investigate the static and dynamic properties of the model, using both the AA and numerical diagonalization. Our analysis reveals a multitude of exceptional points (EPs) that are closely connected with the exactly solvable points in the Hermitian counterpart of the model. Intriguingly, these EPs vanish and revive depending on the light-matter coupling strength. Furthermore, we discuss the time evolution of physical observables under the non-Hermitian Hamiltonian. Rich and exotic behaviors are observed in both strong and ultrastrong coupling regimes. Our work extends the theory of PT\mathcal{PT}PT symmetry into the full quantum light-matter interaction regime and provides insights that can be readily enlarged to a broad class of quantum optical systems.